Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A bound on chaos

Juan Maldacena, Stephen H. Shenker, Douglas Stanford

TL;DR

This paper proposes a universal bound on the growth of chaos in thermal quantum systems with many degrees of freedom, stating that the Lyapunov exponent satisfies $\lambda_L \le \frac{2\pi}{\beta}$ (i.e., $2\pi k_B T/\hbar$). The authors develop a two-part argument: a mathematical bound on a suitably defined analytic correlator in a thermal strip, and a physical justification that the relevant correlators satisfy the required factorization and analyticity properties in broad classes of chaotic systems. They provide evidence from large-N holographic theories, robustness to higher-derivative bulk corrections, Rindler-space arguments, and semiclassical models, and they discuss caveats, including systems where factorization fails. The work suggests a deep connection between causality, analyticity, and gravitational scattering in determining the ultimate speed of quantum information scrambling, with Einstein gravity potentially saturating the bound in suitable theories.

Abstract

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $λ_L \le 2 πk_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

A bound on chaos

TL;DR

This paper proposes a universal bound on the growth of chaos in thermal quantum systems with many degrees of freedom, stating that the Lyapunov exponent satisfies (i.e., ). The authors develop a two-part argument: a mathematical bound on a suitably defined analytic correlator in a thermal strip, and a physical justification that the relevant correlators satisfy the required factorization and analyticity properties in broad classes of chaotic systems. They provide evidence from large-N holographic theories, robustness to higher-derivative bulk corrections, Rindler-space arguments, and semiclassical models, and they discuss caveats, including systems where factorization fails. The work suggests a deep connection between causality, analyticity, and gravitational scattering in determining the ultimate speed of quantum information scrambling, with Einstein gravity potentially saturating the bound in suitable theories.

Abstract

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent . We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

Paper Structure

This paper contains 14 sections, 35 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conjecture
  3. Motivation for the conjecture
  4. Argument
  5. A mathematical result
  6. Deriving the bound
  7. Examples
  8. Large $N$ systems
  9. Extended local systems
  10. Cases where there is no bound
  11. Rindler space and the scattering bound
  12. Semiclassical billiards
  13. Concluding remarks
  14. Rindler space and the scattering bound

Figures (2)

  • Figure 1: $F(t)$ is the correlation function of operators arranged on the thermal circle as shown at left. The folds indicate the Lorentzian time evolution to produce $W(t)$. At complex time, $F(t+i\tau)$ is given by rotating one of the pairs of operators by angle $2\pi\tau/\beta$. The center panel corresponds to $|\tau| < \beta/4$ and the right panel corresponds to $\tau = \beta/4$.
  • Figure 2: Consider a two dimensional Minkowski space, with its right and left Rindler wedges. We insert two operators in the right Rindler wedge and two on the left. Then we act on $W_L$ and $W_R$ by the Rindler time translation generator, which is a boost around the origin. This translates $W_R$ upwards in time and $W_L$ downwards in time, as shown by the arrows.